The frontman, who turns 31 this week, has impressed for the Saints this season after helping them to Premier League promotion last year.
Lambert, who has netted 11 goals this term to make him the highest-scoring Englishman in the top-flight, is out of contract at the end of next season.
His form this term is thought to have alerted other clubs, with West Ham reported to have made an enquiry over the striker in last month's transfer window.
But Lambert says he has no intention of leaving St Mary's stadium and wants to see out his career there.
He told The Sun: "My family are happy here and I am very happy.
"It has been a special three and a half years and I would love to finish my career at Southampton."
Lambert, whose form with the Saints has seen him tipped for a late England call-up, added: "The club's ambitions are very high.
"Southampton is going to be an exciting place over the next few years."